Riemann zeta function: Zeros (formula 10.01.30.0006)

Zeta

$Mathematica Notation$

Zeta Functions and Polylogarithms

Zeta[s]

Zeros

Sums over zeros

http://functions.wolfram.com/10.01.30.0006.01

Input Form

Subscript[Î–,6]ï±2*StieltjesGamma[1]^3 + 6*EulerGamma^4* StieltjesGamma[1] + StieltjesGamma[3]* StieltjesGamma[1] + EulerGamma^6 + (3*StieltjesGamma[2]^2)/4 + 3*EulerGamma^3* StieltjesGamma[2] + EulerGamma^2* (9*StieltjesGamma[1]^2 + StieltjesGamma[3]) + (1/4)*EulerGamma*(24*StieltjesGamma[1]* StieltjesGamma[2] + StieltjesGamma[4]) + StieltjesGamma[5]/20 - Pi^6/960 + 1/; Subscript[Î–,6]ï±Limit[Sum[1/Subscript[Ï,k]^6,Abs[Subscript[Ï,k]<T]], Tï”¢Infinity]&&Zeta[Subscript[Ï,k]]ï±0&& Re[Subscript[Ï,k]]â‰ 0

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[SubscriptBox["\[CapitalZeta]", "6"], "\[Equal]", RowBox[List["1", "+", SuperscriptBox["EulerGamma", "6"], "-", FractionBox[SuperscriptBox["\[Pi]", "6"], "960"], "+", RowBox[List["6", " ", SuperscriptBox["EulerGamma", "4"], " ", RowBox[List["StieltjesGamma", "[", "1", "]"]]]], "+", RowBox[List["2", " ", SuperscriptBox[RowBox[List["StieltjesGamma", "[", "1", "]"]], "3"]]], "+", RowBox[List["3", " ", SuperscriptBox["EulerGamma", "3"], " ", RowBox[List["StieltjesGamma", "[", "2", "]"]]]], "+", FractionBox[RowBox[List["3", " ", SuperscriptBox[RowBox[List["StieltjesGamma", "[", "2", "]"]], "2"]]], "4"], "+", RowBox[List[RowBox[List["StieltjesGamma", "[", "1", "]"]], " ", RowBox[List["StieltjesGamma", "[", "3", "]"]]]], "+", RowBox[List[SuperscriptBox["EulerGamma", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["9", " ", SuperscriptBox[RowBox[List["StieltjesGamma", "[", "1", "]"]], "2"]]], "+", RowBox[List["StieltjesGamma", "[", "3", "]"]]]], ")"]]]], "+", RowBox[List[FractionBox["1", "4"], " ", "EulerGamma", " ", RowBox[List["(", RowBox[List[RowBox[List["24", " ", RowBox[List["StieltjesGamma", "[", "1", "]"]], " ", RowBox[List["StieltjesGamma", "[", "2", "]"]]]], "+", RowBox[List["StieltjesGamma", "[", "4", "]"]]]], ")"]]]], "+", FractionBox[RowBox[List["StieltjesGamma", "[", "5", "]"]], "20"]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["\[CapitalZeta]", "6"], "\[Equal]", RowBox[List["Limit", "[", RowBox[List[RowBox[List[UnderscriptBox["\[Sum]", RowBox[List[RowBox[List["\[LeftBracketingBar]", SubscriptBox["\[Rho]", "k"], "\[RightBracketingBar]"]], "<", "T"]]], FractionBox["1", SubsuperscriptBox["\[Rho]", "k", "6"]]]], ",", RowBox[List["T", "\[Rule]", "\[Infinity]"]]]], "]"]]]], "\[And]", RowBox[List[RowBox[List["Zeta", "[", SubscriptBox["\[Rho]", "k"], "]"]], "\[Equal]", "0"]], "\[And]", RowBox[List[RowBox[List["Re", "[", SubscriptBox["\[Rho]", "k"], "]"]], "\[NotEqual]", "0"]]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msub> <mi> Ζ </mi> <mn> 6 </mn> </msub> <mo> ⩵ </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msubsup> <semantics> <mi> γ </mi> <annotation encoding='Mathematica'> TagBox["\[Gamma]", StieltjesGamma] </annotation> </semantics> <mn> 1 </mn> <mn> 3 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mn> 6 </mn> <mo> ⁢ </mo> <msup> <semantics> <mi> ℽ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubledGamma]", Function[EulerGamma]] </annotation> </semantics> <mn> 4 </mn> </msup> <mo> ⁢ </mo> <msub> <semantics> <mi> γ </mi> <annotation encoding='Mathematica'> TagBox["\[Gamma]", StieltjesGamma] </annotation> </semantics> <mn> 1 </mn> </msub> </mrow> <mo> + </mo> <mrow> <msub> <semantics> <mi> γ </mi> <annotation encoding='Mathematica'> TagBox["\[Gamma]", StieltjesGamma] </annotation> </semantics> <mn> 3 </mn> </msub> <mo> ⁢ </mo> <msub> <semantics> <mi> γ </mi> <annotation encoding='Mathematica'> TagBox["\[Gamma]", StieltjesGamma] </annotation> </semantics> <mn> 1 </mn> </msub> </mrow> <mo> + </mo> <msup> <semantics> <mi> ℽ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubledGamma]", Function[EulerGamma]] </annotation> </semantics> <mn> 6 </mn> </msup> <mo> + </mo> <mfrac> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <msubsup> <semantics> <mi> γ </mi> <annotation encoding='Mathematica'> TagBox["\[Gamma]", StieltjesGamma] </annotation> </semantics> <mn> 2 </mn> <mn> 2 </mn> </msubsup> </mrow> <mn> 4 </mn> </mfrac> <mo> + </mo> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <msup> <semantics> <mi> ℽ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubledGamma]", Function[EulerGamma]] </annotation> </semantics> <mn> 3 </mn> </msup> <mo> ⁢ </mo> <msub> <semantics> <mi> γ </mi> <annotation encoding='Mathematica'> TagBox["\[Gamma]", StieltjesGamma] </annotation> </semantics> <mn> 2 </mn> </msub> </mrow> <mo> + </mo> <mrow> <msup> <semantics> <mi> ℽ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubledGamma]", Function[EulerGamma]] </annotation> </semantics> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 9 </mn> <mo> ⁢ </mo> <msubsup> <semantics> <mi> γ </mi> <annotation encoding='Mathematica'> TagBox["\[Gamma]", StieltjesGamma] </annotation> </semantics> <mn> 1 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> + </mo> <msub> <semantics> <mi> γ </mi> <annotation encoding='Mathematica'> TagBox["\[Gamma]", StieltjesGamma] </annotation> </semantics> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> ⁢ </mo> <semantics> <mi> ℽ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubledGamma]", Function[EulerGamma]] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 24 </mn> <mo> ⁢ </mo> <msub> <semantics> <mi> γ </mi> <annotation encoding='Mathematica'> TagBox["\[Gamma]", StieltjesGamma] </annotation> </semantics> <mn> 1 </mn> </msub> <mo> ⁢ </mo> <msub> <semantics> <mi> γ </mi> <annotation encoding='Mathematica'> TagBox["\[Gamma]", StieltjesGamma] </annotation> </semantics> <mn> 2 </mn> </msub> </mrow> <mo> + </mo> <msub> <semantics> <mi> γ </mi> <annotation encoding='Mathematica'> TagBox["\[Gamma]", StieltjesGamma] </annotation> </semantics> <mn> 4 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mfrac> <msub> <semantics> <mi> γ </mi> <annotation encoding='Mathematica'> TagBox["\[Gamma]", StieltjesGamma] </annotation> </semantics> <mn> 5 </mn> </msub> <mn> 20 </mn> </mfrac> <mo> - </mo> <mfrac> <msup> <mi> π </mi> <mn> 6 </mn> </msup> <mn> 960 </mn> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <msub> <mi> Ζ </mi> <mn> 6 </mn> </msub> <mo> ⩵ </mo> <mrow> <munder> <mi> lim </mi> <mrow> <mi> T </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mi> ∞ </mi> </mrow> </munder> <mo> ⁢ </mo> <mtext>   </mtext> <mrow> <mo> ( </mo> <mrow> <munder> <mo> ∑ </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <msub> <mi> ρ </mi> <mi> k </mi> </msub> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> = </mo> <mn> 1 </mn> </mrow> </munder> <mfrac> <mn> 1 </mn> <msubsup> <mi> ρ </mi> <mi> k </mi> <mn> 6 </mn> </msubsup> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ∧ </mo> <mrow> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> ρ </mi> <mi> k </mi> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox[SubscriptBox["\[Rho]", "k"], Rule[Editable, True]], ")"]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> <mo> ⩵ </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> ρ </mi> <mi> k </mi> </msub> <mo> ) </mo> </mrow> <mo> ≠ </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> FormBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> Ζ </ms> <ms> 6 </ms> </apply> <ms> ⩵ </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <apply> <ci> SubsuperscriptBox </ci> <apply> <ci> TagBox </ci> <ms> γ </ms> <ci> StieltjesGamma </ci> </apply> <ms> 1 </ms> <ms> 3 </ms> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <ms> 6 </ms> <apply> <ci> SuperscriptBox </ci> <apply> <ci> TagBox </ci> <ms> ℽ </ms> <apply> <ci> Function </ci> <eulergamma /> </apply> </apply> <ms> 4 </ms> </apply> <apply> <ci> SubscriptBox </ci> <apply> <ci> TagBox </ci> <ms> γ </ms> <ci> StieltjesGamma </ci> </apply> <ms> 1 </ms> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <apply> <ci> TagBox </ci> <ms> γ </ms> <ci> StieltjesGamma </ci> </apply> <ms> 3 </ms> </apply> <apply> <ci> SubscriptBox </ci> <apply> <ci> TagBox </ci> <ms> γ </ms> <ci> StieltjesGamma </ci> </apply> <ms> 1 </ms> </apply> </list> </apply> <ms> + </ms> <apply> <ci> SuperscriptBox </ci> <apply> <ci> TagBox </ci> <ms> ℽ </ms> <apply> <ci> Function </ci> <eulergamma /> </apply> </apply> <ms> 6 </ms> </apply> <ms> + </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> 3 </ms> <apply> <ci> SubsuperscriptBox </ci> <apply> <ci> TagBox </ci> <ms> γ </ms> <ci> StieltjesGamma </ci> </apply> <ms> 2 </ms> <ms> 2 </ms> </apply> </list> </apply> <ms> 4 </ms> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <ms> 3 </ms> <apply> <ci> SuperscriptBox </ci> <apply> <ci> TagBox </ci> <ms> ℽ </ms> <apply> <ci> Function </ci> <eulergamma /> </apply> </apply> <ms> 3 </ms> </apply> <apply> <ci> SubscriptBox </ci> <apply> <ci> TagBox </ci> <ms> γ </ms> <ci> StieltjesGamma </ci> </apply> <ms> 2 </ms> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <apply> <ci> TagBox </ci> <ms> ℽ </ms> <apply> <ci> Function </ci> <eulergamma /> </apply> </apply> <ms> 2 </ms> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 9 </ms> <apply> <ci> SubsuperscriptBox </ci> <apply> <ci> TagBox </ci> <ms> γ </ms> <ci> StieltjesGamma </ci> </apply> <ms> 1 </ms> <ms> 2 </ms> </apply> </list> </apply> <ms> + </ms> <apply> <ci> SubscriptBox </ci> <apply> <ci> TagBox </ci> <ms> γ </ms> <ci> StieltjesGamma </ci> </apply> <ms> 3 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 4 </ms> </apply> <apply> <ci> TagBox </ci> <ms> ℽ </ms> <apply> <ci> Function </ci> <eulergamma /> </apply> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 24 </ms> <apply> <ci> SubscriptBox </ci> <apply> <ci> TagBox </ci> <ms> γ </ms> <ci> StieltjesGamma </ci> </apply> <ms> 1 </ms> </apply> <apply> <ci> SubscriptBox </ci> <apply> <ci> TagBox </ci> <ms> γ </ms> <ci> StieltjesGamma </ci> </apply> <ms> 2 </ms> </apply> </list> </apply> <ms> + </ms> <apply> <ci> SubscriptBox </ci> <apply> <ci> TagBox </ci> <ms> γ </ms> <ci> StieltjesGamma </ci> </apply> <ms> 4 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> + </ms> <apply> <ci> FractionBox </ci> <apply> <ci> SubscriptBox </ci> <apply> <ci> TagBox </ci> <ms> γ </ms> <ci> StieltjesGamma </ci> </apply> <ms> 5 </ms> </apply> <ms> 20 </ms> </apply> <ms> - </ms> <apply> <ci> FractionBox </ci> <apply> <ci> SuperscriptBox </ci> <ms> π </ms> <ms> 6 </ms> </apply> <ms> 960 </ms> </apply> <ms> + </ms> <ms> 1 </ms> </list> </apply> </list> </apply> <ms> /; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> Ζ </ms> <ms> 6 </ms> </apply> <ms> ⩵ </ms> <apply> <ci> RowBox </ci> <list> <ms> Limit </ms> <ms> [ </ms> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderscriptBox </ci> <apply> <ci> ErrorBox </ci> <ms> ∑ </ms> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms>  </ms> <apply> <ci> SubscriptBox </ci> <ms> ρ </ms> <ms> k </ms> </apply> <ms>  </ms> </list> </apply> <ms> = </ms> <ms> 1 </ms> </list> </apply> </apply> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <apply> <ci> SubsuperscriptBox </ci> <ms> ρ </ms> <ms> k </ms> <ms> 6 </ms> </apply> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <ms> T </ms> <ms> -> </ms> <ms> ∞ </ms> </list> </apply> <ms> ] </ms> </list> </apply> </list> </apply> <ms> ∧ </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ζ </ms> <ms> ( </ms> <apply> <ci> TagBox </ci> <apply> <ci> SubscriptBox </ci> <ms> ρ </ms> <ms> k </ms> </apply> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> InterpretTemplate </ci> <apply> <ci> Function </ci> <ci> BoxForm`e$ </ci> <apply> <ci> Zeta </ci> <ci> BoxForm`e$ </ci> </apply> </apply> </apply> </apply> <ms> ⩵ </ms> <ms> 0 </ms> </list> </apply> <ms> ∧ </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> Re </ms> <ms> ( </ms> <apply> <ci> SubscriptBox </ci> <ms> ρ </ms> <ms> k </ms> </apply> <ms> ) </ms> </list> </apply> <ms> ≠ </ms> <ms> 0 </ms> </list> </apply> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>

Date Added to functions.wolfram.com (modification date)

2003-08-21

Zeta[s,a]